Induced connections on $S^ 1$-bundles over Riemannian manifolds

Abstract:

Let $(V,g)$ and $(W,h)$ be Riemannian manifolds and consider two ${S^1}$-bundles $X \to V$ and $Y \to W$ with connections $\Gamma$ on $X$ and $\nabla$ on $Y$ respectively. We study maps $X \to Y$ which induce both connections and metrics. Our study relies on Nash’s implicit function theorem for infinitesimally invertible differential operators. We show, for the case when $Y \to W = {\mathbf {C}}{P^q}$ is the Hopf bundle, that if $2q \geq n(n + 1)/2 + 3n$ then there exists a nonempty open subset in the space of ${C^\infty }$-pairs $(g,\Gamma )$ on $V$ which can be induced from $(h,\nabla )$ on ${\mathbf {C}}{P^Q}$.